USB Proceedings
2015 IEEE International Electric
Machines and Drives Conference
(IEMDC)
Coeur d'Alene Resort
Coeur d'Alene, ID, U.S.A.
11 - 13 May, 2015
Sponsored by
The Institute of Electrical and Electronics Engineers (IEEE)
IEEE Industrial Electronics Society (IES)
Co-sponsored by
IEEE Power and Energy Society (PES)
IEEE Industry Applications Society (IAS)
IEEE Power Electronics Society (PES)
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IEEE Catalog Number: CFP15EMD-USB
ISBN: 978-1-4799-7940-0
2
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Application of sinusoidal field pole
in a permanent magnet synchronous machine
to improve the acoustic behavior
considering the MTPA and MTPV operation area
Aryanti Kusuma Putri, Sebastian Rick, David Franck and Kay Hameyer
Institute of Electrical Machines
RWTH Aachen University
Email: aryanti.putri@iem.rwth-aachen.de
Abstract—In this paper, various approaches to improve the
acoustic behavior of a single layer interior permanent magnet
synchronous machine (IPMSM) are evaluated. The studied machine is designed for the application in electric vehicles and has
a maximum output power of 53.6 kW. The acoustic noise of the
electrical machine is mainly generated due to vibration of the
stator yoke, which is caused by the radial forces acting on the
stator’s teeth. The radial force is directly correlated with the flux
density in the air gap. Through modification of the outer frame
of the rotor, the composition of the air gap field, which consists
of the fundamental wave and its harmonics, can be modified.
The distribution of the radial force density on the surface of the
stator teeth will be affected and the dominant order of the radial
force can be reduced selectively, which will improve its acoustic
behavior.
Index Terms—PMSM, acoustic and noise of electrical machines,
air gap flux, sinusoidal field pole, rotor pole shaping, torque
harmonics, force densities, electric vehicles
I.
I NTRODUCTION
The role of electric vehicles in individual transportation
has become significant in the recent years. Although electrical
drives are known for the low noise emission when compared
to combustion engine, significant acoustic problems originated
from single tones attract particular attention. Combustion engines, in contrast to the electric motor, generate broadband
noise. Noise from electric motor has a single tone characteristic
and lies within the sensitive hearing range of humans, which
is generally between 2-5 kHz [1]. The sound radiation is
strongly dependent on the housing and the installation of the
machine. For a feasible acoustic evaluation, these aspects have
to be considered. Therefore, structural dynamic and acoustic
simulations have to be performed [2]. However, in the case that
the dominant frequencies of the acoustic radiation are known, it
is sufficient to analyze the exciting forces for the improvement
of the electrical drive.
The major sources of noise in electrical machines are the
periodic radial and tangential forces waves acting on stator and
rotor respectively. The radial forces acting on the stator lead
to the deformation of the housing. The tangential forces acting
on the rotor leads to torque and torque pulsation. Both result
in unwanted vibration of the machine parts, e.g. shaft, gearing,
and differential, which is followed by acoustic radiation. Since
the sum of the torque depends on the integration of tangential
978-1-4799-7940-0/15/$31.00 ©2015 IEEE
1359
forces and the axial length, the pulsation can be reduced by
skewing the stator or rotor. The sum of the radial forces on the
stator circumference in a time step is null. In this case, the local
force densities distribution have to be analyzed. The local force
densities are related to the flux density waves on the air gap.
As example in an induction machine with a squirrel cage rotor,
these harmonics can be reduced by choosing specific numbers
of stator and rotor slots [3]. In a salient pole synchronous
machine, the flux density harmonics can be reduced by altering
the pole shoe shape of the rotor [4].
In this paper, a systematic approach to minimize the
flux density harmonics of the air gap to reduce the acoustic radiation of an interior permanent magnet synchronous
machine (IPMSM) is presented. The focus of this work lies
on the improvement of the electromagnetic circuit through
forming the rotor’s surface and thus the shape of the airgap.
The stator, housing and other machine components are not
altered, so the time consuming structural dynamic and acoustic
simulations can be avoided. The resulting torque harmonics
(temporal), which represent the sum of tangential force waves
(spatial), and radial force density waves (temporal and spatial)
are analyzed. The velocity on the stator surface caused by
the radial force density waves acting on the stator teeth is
evaluated. The proposed method is applied in a nonskewed
single layer interior permanent magnet synchronous machine
(IPMSM). The simulation and machine parameters are listed in
Table I. The results are analyzed for the entire operating area,
which is defined through maximal torque per ampere (MTPA)
in base speed and maximal through per voltage (MTPV) in
field weakening operating points [5].
TABLE I: Simulation and Machine Parameters of the IPMSM
Stator outer radius rstator,o
Stator inner radius rstator,i
Air gap length δ
Axial length lF e
Number of Poles / Slots 2p/N
Battery voltage Udc
Maximum current (peak value) Imax
Rated torque MN
Rated speed nN
Rated power PN
90 mm
61 mm
0.7 mm
120 mm
12 / 36
300 V
300 A
155 Nm
3300 min−1
53.6 kW
II.
M ETHODOLOGY
A. Analysis of the Radial Force Density Waves
The radial force density waves acting on the stator teeth are
the major reason of the vibration of the stator’s teeth, which
leads to the deformation of the stator yoke. These force density
waves can be traced back to the harmonics of flux densities,
which originate from the rotor or stator. Generally, the flux
density harmonics are caused by permanent magnets, stator
slots and winding, and other effects such as saturation and
eccentricity. The radial force density waves σrad (α, t) can be
calculated with the simplified equation [6]:
Brad (α, t)2
(1)
2 µ0
B1 (α, t)2 + 2 B1 (α, t) B2 (α, t) + B2 (α, t)2
,
=
2 µ0
σrad (α, t)=
whereas α and t describe the position in the airgap and point
in time. µ0 , Brad , B1 and B2 are magnetic constant, sum
of radial flux density in the airgap, radial flux density from
stator and rotor respectively. After [7], the force density waves
described in (1) can be decomposed according to its source of
flux density waves. The force densities with r-th spatial order
σrad,r , which occur due to the flux density harmonics of stator
or rotor in combination with itselves, are described with the
equation:
Brad,ν/µ (α, t)2
σrad,r (α, t) =
(2)
2 µ0
The force densities with r-th spatial order σrad,r , which occur
due the combination of the flux density harmonics of stator
and rotor, are described with the equation:
2 Brad,ν (α, t) Brad,µ (α, t)
σrad,r (α, t) =
.
2 µ0
(3)
Brad,ν and Brad,µ are radial flux density ν-th order from stator
and µ-th order from rotor (spatial) respectively. The deformation’s amplitude r-th spatial order Yr of the stator yoke, which
is caused by these force densities, can be estimated with the
equations [3]:
rstator,i Nyoke
σrad,0
EF e hyoke
4
rstator,i lF e
Y1 =
σrad,1
3 EF e ( dshaf t )4 Lshaf t
Lshaf t
√
2 3 Nyoke 1
)2 σrad,r .
Yr>=2 = Y0 (
hyoke r2 − 1
Y0 = −
(4)
dshaf t and Lshaf t are the diameter of the shaft and the distance
of the bearings respectively. Assuming that the deformation
of the stator is an oscillation in stationary state, the surface
velocities Ẏ can be calculated with:
Ẏr = 2 π fmech s Yr ,
(5)
whereas fmech and s are the mechanical rotation frequency
and temporal order respectively. The deformation of the stator
yoke depending on spatial order r is illustrated in Fig. 1. A
deformation with spatial order r rotates with the frequency
fmech ∗ s and produces a tone with the same frequency. It
is shown in (4), that the amplitude of the stator deformation
Yr decreases with 1/r4 . Therefore, the relevant radial force
1360
r=0
r=1
r=2
r=3
r=4
r=5
Fig. 1: The deformation of the stator yoke depending on spatial order r.
densities, i.e. force density waves with low spatial order,
correspond to the dominant amplitude of the stator deformation
and thus its surface velocity. Hence, it is sufficient to analyze
the amplitude of stator deformation or the surface velocity
to determine the force densities, which are prominent to the
acoustic radiation.
Through convolution of a radial force density wave, the
time and spatial orders of the flux densities, which have major
contribution to the radial force density, can be determined
[8]. The terms in (1) can be two-dimensional (2D) Fourier
transformed and the equation can be rewritten in:
σDF T 2 (s, r) =
BDF T 2 (ν/µ, s) ∗ BDF T 2 (ν/µ, s)
.
2 µ0
(6)
The Fourier transformed force and flux densities (σDF T 2 and
BDF T 2 respectively) are dependent on temporal order s and
spatial orders:
•
of the force density waves r,
•
of the flux density waves originated from stator ν,
•
of the flux density waves originated from rotor µ.
To determine the origin of the flux density waves, finite element simulations (FE-simulations) of rotor with magnetic ideal
slotless stator or stator with magnetic ideal rotor (µr → ∞ or
Neumann boundary condition) need to be performed. As a
result, the source of flux density harmonics in the air gap can
be separated.
B. Analysis of the Tangential Force Density Waves
According to [9], the sum of the forces Fk on every node
k (nodal force) multiplied by its distance to the origin of the
rotor rk represent the torque T in one point in time. It is
described with the equation:
T = Σ r k × Fk .
(7)
Due to the cross product, only the tangential nodal forces
have influence on the torque. Through implementation of
(7) for multiple time steps up to one rotation of the rotor,
the mean torque and torque harmonics can be calculated.
Through analyzing the time-dependent torque, the effect of
the tangential force densities wave can be studied.
The next step is to determine the radial force density waves,
which have major contribution to the acoustic radiation. It can
be determined through measurement of sound pressure level
(SPL) or estimated through simulation. By measuring the SPL,
the dominant temporal orders of the acoustic radiation can be
determined. The lowest spatial order of these temporal orders
should be chosen for further study. Without measurement
results, the dominant excitation of the acoustic radiation can
only be estimated. Generally, the pair of dominant torque harmonics (temporal order) with the lowest spatial order should be
chosen. Furthermore, the dominant force density waves with
low temporal and spatial order should be analyzed.
The chosen force density waves are convoluted to identify
the flux density waves, which have major contribution to these
force densities. Depending on the result, the rotor and stator
geometry or its winding can be altered to minimize the flux
density harmonics in the air gap. This paper focuses on the
flux density originated from the rotor and the alteration is
performed on the outer surface of the rotor.
III.
A LTERATION OF ROTOR ’ S S URFACE
In this section two approaches to reduce flux density
harmonics by alternating the outer surface of the rotor will
be introduced. The first approach is using the sinusoidal rotor
field poles calculated with inverse cosine formula. The second
approach is the notching of rotor’s surface in d- or q-axis. The
method to evaluate the influence of rotor’s alteration on the air
gap flux density is described at the end of the section.
Nodal Force [N]
35
25
25
20
15
10
5
0
Fig. 2: The average of nodal forces between two neighboring nodes.
1361
q-axis
d-axis
60
rrotor,o (β)
to r
,o
50
40
rro
The first step of the approach is to perform FE-simulations
to determine the nodal forces on the air gap side of the
stator’s surface. The force density waves acting on the stator
are calculated through averaging two neighboring nodal forces
(Fig. 2) and dividing its value by the distance between the two
nodes and the axial length of the machine. The force densities
should be decomposed in radial and tangential components.
Through 2D Fourier transformation, the force densities can be
decomposed in temporal and spatial order. As shown in Fig. 2,
the average of nodal forces and thus the force density waves
point mainly in radial direction. The value of the force densities
in radial direction is greater than its value in the tangential
direction. For this reason, it is sufficient to analyze only the
radial force density waves to prognose the acoustic behavior
of the machine. In case the acoustic radiation is excited by the
torque harmonics, the correlated radial force density waves
(temporal order) should be analyzed.
y-coordinate [mm]
C. Implementation
β
rrotor,o,min
30
20
10
r ro
0
0
to
i
r,
10
20
30
x-coordinate [mm]
Fig. 3: Illustration of sinusoidal field pole and its parameters.
A. Sinusoidal Rotor Field Poles
Through methods introduced in [4] the harmonics of air
gap flux densities can be reduced and sinusoidal rotor field
can be reached. The alteration of the rotor’s surface is defined
with the equations:
δ(β) =
δd
cos( τπp β)
rrotor,o (β) = rrotor,o + δd − δ(β),
(8)
(9)
whereas δd is air gap length in the d-axis, τp is the pole pitch, β
is the angle relative to d-axis of the rotor, δ(β) and rrotor,o (β)
are the air gap length and rotor outer radius dependent on β
respectively. In Fig. 3 a half of a rotor pole is shown and
the parameters in (8) and (9) are illustrated. According to
the previous equations, the rotor outside radius should follow
the dashed contour pointed by rrotor,o (β). This parameter is
limited with the rotor minimal outside radius rrotor,o,min and
hence the air gap length in the q-axis δq . The ratio of the air
gap length in q- and d-axis δq/δd is varied and its influence on
the flux density composition is evaluated.
B. Notching of Rotor’s Surface
As an alternative to the previous section, the modifications
of rotor surface on the d- and q-axis are considered. The
cogging torque of an IPMSM can be reduced by notching
the rotor surface in the d-axis [10] or q-axis [11]. The notch
parameters are shown in Fig. 4, which are notch angle to dor q-axis αnotch and notch depth dnotch . The influence of
both parameters on the flux density harmonics is evaluated
separately.
IV.
dnotch
A PPLICATION ON THE IPMSM
The methods, which are explained in the previous sections,
are applied to the examined IPMSM. First of all, the relevant
operating points for the analysis have to be chosen. To examine the forces acting on the rotor, the torque harmonics
have to be analyzed. The first dominant torque harmonic of
the examined IPMSM is the 36th harmonic, which is the
first slot harmonics. In Fig. 5 the characteristic torque-speeddiagram of this harmonic is shown. From the diagram could
be concluded, that the maximum of this harmonic is located
at the maximum torque in the base speed area. The amplitude
is 30 Nm, which represents 20 % of the average torque. The
operating point between base speed and field weakening area
(n = 3300 min−1 and M = 155 Nm) at rated operating point
is chosen for further examination.
Torque harmonic [Nm], temporal order s = 36
30
200
αnotch
(a)
dnotch
180
25
Torque M [Nm]
160
αnotch
140
20
120
100
15
80
10
60
(b)
40
Fig. 4: Notch on d-axis (a) or q-axis (b).
0
0
C. Evaluation of Flux Density Waves
The flux density harmonics originated from rotor are evaluated. The stator is treated as an ideal slotless stator, which
is implemented in the FE-simulation by applying neumann
boundary condition on the stator inner radius. With this
method, there is only one time step required for the analysis
of the flux density, since in this case the flux density is
not time-dependent. The flux density in the middle of the
airgap rδ,mid = 60, 65 mm is sampled and Fourier transformed
(spatial). The total harmonic distortion (THD) is evaluated with
the equation:
v
uX
2
u
Brad,µ
u
u µ>1
T HD =u
.
(10)
uX
2
t
Brad,µ
µ
The objective of the rotor’s surface alteration is to reduce the
spatial flux density harmonics without reducing the fundamental wave significantly. In addition, the lower harmonics of the
flux density have a major effect on the parasitic effects. Thus,
the reduction of lower harmonics has a higher priority than
recution of the higher harmonics. Therefore, the evaluation
criteria of the flux density harmonics are:
•
the total harmonic distortion,
•
the value of the flux density’s fundamental wave,
•
5
20
the value of the flux density’s harmonics µ = 3, 5, 7.
1362
5000
10000
Speed n [min−1 ]
15000
0
Fig. 5: Characteristic torque-speed-diagram of the 36th harmonic of torque.
To examine the radial forces, which cause the deformation
of the housing, the radial force density waves have to be
analyzed. The low spatial orders r with highest value of
the force densities have to be chosen. According to [7],
the dominant force density orders of the examined machine
are (temporal s, spatial r): (12, 12), (24, -12) and (36, 0). In
Fig. 6 the characteristic torque-speed-diagramm for the radial
force density order (36, 0) is shown. The operating points with
the highest force density value are chosen for further examination. The chosen operating points for the three aforementioned
force density orders are listed in Table II. Calculation of
surface velocities will be performed on these operating points.
In Fig. 7 the calculated surface velocities on the stator
surface are shown. In all studied operationg points, the surface
velocity reaches its maximum at temporal order s = 36 and
TABLE II: Operating Points for Calculation of Surface Velocities
Force density orders
(s, r) [kN/m2 ]
σDF T 2 (12, 12) = 200
σDF T 2 (24, −12) = 22.2
σDF T 2 (36, 0) = 36
Speed
n [min−1 ]
Torque
M [Nm]
Current
Ief f [A]
Control angle
ψ [◦ ]
3300
8000
7400
155
30
82.5
210
95
200
-30
-70
-70
2
35
-Re
180
1
3
30
4
Torque M [Nm]
160
5
140
25
120
20
6
7
100
80
15
60
10
8
9
10
K
LM
N
40
5
20
0
0
5000
10000
Speed n [min−1 ]
15000
-Im
0
Fig. 8: Space vector convolution of the force density order (36,0)
at operating point n = 7400 min−1 and M = 82.5 Nm.
Fig. 6: Characteristic torque-speed-diagram of the force density wave order
(temporal s, spatial r) = (36, 0).
spatial order r = 0. The next dominant orders are (s, r) =
(72, 0) and (s, r) = (12, 12). It is shown, that the values of
the force densities are not directly correlated with the surface
velocities, and therefore the acoustic radiation. The acoustic radiation is strongly dependent on the temporal and spatial order
of the force excitations. The frequencies of the noise excited
from the force density order (s, r) = (36, 0) between the speed
n = 3300 − 8000 min−1 lie between f = 1980 − 4800 Hz.
These frequencies are located in the area of the human
sensitive hearing range. To improve the acoustic behavior of
the IPMSM this force density order have to be reduced. The
operating point n = 7400 min−1 and M = 82.5 Nm shows the
highest surface velocity order (36, 0), when compared to the
other operating points. This radial force density order in the
correspondent operating point will be convoluted to determine
the contributing flux densities.
Spatial order r
–24
–12
2
1.5
0
1
12
0.5
24
0
0
–24
Spatial order r
Surface Velocity [mm/s]
n = 3300 min−1 & M = 155 Nm
36
72
108
Frequency order s
n = 7400 min−1 & M = 82.5 Nm
–12
24
3
2
36
72
Frequency order s
108
Vector
number
1&2
3&4
5&6
7&8
9 & 10
K&L
M&N
Flux Density Pair
Brad (s, r)
Brad (s, r)
6, 6
42, 42
6, -6
30, 30
18, 18
18, -18
54, 54
30, -6
6, -42
42, 6
6, -30
18, -18
54, 18
18, -54
In Fig. 8 the decomposition of the radial force density
order (36, 0) (σDF T 2 (36, 0)) into its constructing force densities (numbered arrows) is shown. The vectors are sorted in
descending order after their absolute value. The pairs of the
flux density orders, which build each force density vectors,
are listed in Table III. The flux densities order (6, 6), (18, 18),
(30, 30), (42, 42) and (54, 54) are originated from rotor. The
normed orders (after polepair p and simulation of a time step)
are 1st , 3rd , 5th , 7th and 9th respectively. The fundamental
wave is the order (6, 6) or the 1st order. The value of the
higher harmonics have to be reduced, to improve the acoustic
behavior. The objective is to alter the rotor flux density of
the IPMSM into a sinusoidal form, as shown in Fig. 9. In the
next subsections, the rotor alteration will be performed and its
effect on the flux density will be examined.
Simulations of rotors with varied air gap ratio δq /δd = 1−3
with an increment of 0.25 are performed. The minimum width
of the bridge between the magnet and rotor outside contour
is set to 0.3 mm. To meet this condition, the position of the
magnets relative to rotor outside contour will be changed and
their geometry are kept constant. The results are shown in
Table IV. It is shown, that the THD of the air gap flux density
decreases significantly until the air gap ratio δq /δd = 2 is
reached. The harmonics sink remarkably until the air gap ratio
δq /δd = 1.75. The fundamental wave of the flux density
increases and reaches its maximum between δq /δd = 2 − 2.75.
In Fig. 9 the spatial distribution of the flux densities for
δq /δd = 2 & 2.25 are shown. They are compared to the flux
1
0
TABLE III: Contributing Flux Density Orders
A. Sinusoidal Rotor Field Poles
0
12
σDF T 2 (36, 0)
Force density [kN/m2 ], (s, r) = (36, 0)
200
0
Fig. 7: The excited surface velocities at two operating points.
1363
1
0.8
0.8
0.6
Flux density B [T]
Flux density B [T]
0.6
0.4
0.2
0
-0.2
0.2
0
-0.2
-0.4
Original
δq /δd = 2
δq /δd = 2.25
Sine function
20
30
40
50
60
Mechanical angle [◦ ]
-0.6
-0.8
-1
0.4
0
10
-0.4
-0.6
-0.8
0
10
Original
Notch in d-axis
Notch in q-axis
Sine function
20
30
40
50
60
Mechanical angle [◦ ]
Fig. 9: The air gap flux density of IPMSM rotor with and without sinusoidal
field poles.
Fig. 10: The air gap flux density of IPMSM rotor
with and without notch in d- or q-axis.
TABLE IV: Variation of air gap ratio δq /δd
TABLE V: Variation of Notch Parameters αnotch & dnotch in d-Axis
Air gap ratio
δq /δd
THD
[%]
1
1 (Original)
1.25
1.5
1.75
2
2.25
2.5
2.75
3
24.38
17.73
11.81
8.54
6.91
6.42
6.23
5.98
5.53
0.75
0.77
0.80
0.84
0.86
0.86
0.86
0.86
0.85
Flux density order [T]
3
5
7
0.15
0.09
0.04
0.01
0.01
0.02
0.02
0.02
0.02
0.01
0.02
0.01
0
0.02
0.02
0.03
0.03
0.02
0.06
0.05
0.03
0.03
0.03
0.03
0.03
0.03
0.02
9
0.07
0.07
0.06
0.04
0.03
0.03
0.02
0.02
0.02
density of the original IPMSM and a sine function. It is shown
in the distribution, that the flux densities sampled at ±10◦
from d-axis follow the sine function thoroughly. The difference
between the two sinusoidal poles are insignificant. Due to the
wider range of the similarity to the sine function, the rotor with
air gap ratio δq /δd = 2.25 is chosen for further examination.
B. Notching of Rotor’s Surface
The variation of the notch parameters (angle αnotch and
depth dnotch ) in d- and q-axis are performed. The minimum
width of the bridges is kept at 0.3 mm and the magnets’
position isn’t changed. In Fig. 10 the spatial distribution of the
flux densities with notch in d- or q-axis are shown. The notch
in d-axis causes a drop of the flux density in the position of the
notch. The notch in q-axis induces a smoother transition of the
flux density between the poles. The simulation results are listed
in Table V and Table VI. It is shown, that the notch in the daxis induces a lower fundamental wave of flux densities when
compared to a rotor with a notch in q-axis. In both cases, the
THD of the flux densities are higher than the original model.
The value of the 7th harmonic order can be lowered with
both notch’s position. The 9th harmonic order is additionally
lowered in case of a notch in q-axis. The gray rows mark
the geometry with highest reduction the 7th harmonic order,
which has vast contribution to the flux density order (36, 0)
at n = 7400 min−1 and M = 82.5 Nm (Table III). These
geometries are chosen for further examination.
1364
Angle
αnotch [◦ ]
Depth
dnotch [mm]
THD
[%]
1
Flux density order [T]
3
5
7
0.9
0.9
0.25
2
26.63
29.47
0.74
0.74
0.16
0.18
0.02
0.04
0.05
0.04
0.09
0.10
0.5
2
0.5
0.5
25.73
32.48
0.75
0.73
0.15
0.2
0.01
0.06
0.06
0.02
0.07
0.1
1.5
2.5
4
5
0.5
1
1.5
1.75
30.11
38.22
49.44
55.64
0.74
0.72
0.68
0.66
0.19
0.25
0.34
0.39
0.04
0.09
0.13
0.13
0.03
0.01
0.03
0.08
0.01
0.12
0.08
0.04
9
TABLE VI: Variation of Notch Parameters αnotch & dnotch in q-Axis
Angle
αnotch [◦ ]
Depth
dnotch [mm]
THD
[%]
1
Flux density order [T]
3
5
7
0.9
0.9
0.5
4
23.38
23.73
0.75
0.76
0.15
0.17
0.01
0.02
0.06
0.03
0.07
0.04
0.5
1
2.5
2.5
24.25
28.84
0.75
0.81
0.15
0.2
0
0.08
0.06
0.01
0.07
0.02
0.5
1
1.5
2
1.5
2.5
4
3.5
24.37
23.90
26.67
29.78
0.75
0.75
0.79
0.81
0.15
0.15
0.20
0.23
0.01
0
0.07
0.09
0.06
0.05
0.01
0.03
0.07
0.06
0.01
0
9
C. Combination of Presented Approaches
As shown in Fig. 9 and Fig. 10, the combination of sinusoidal field poles and notches in q-axis show the most
promising results. Through the sinusoidal field pole, the flux
density near d-axis follow the shape of a sine function. The
notch in q-axis leads to a smoother transition between the
poles. The chosen air gap ratio δq /δd = 2.25 and notch angle
αnotch = 2◦ are implemented on the rotor. The chosen angle
corresponds to the half of the pole transition angle in the flux
density distribution. The depth of the notch dnotch is 3,5 mm,
which is the maximum depth subjected to the condition of
minimum width of the bridges. In this case, the value of the
fundamental wave is 0.89 T, the THD is 1.69 %, and the value
of the rest of the harmonics are under 0.005 T.
V.
R ESULTS
A. Analysis of an Operating Point
Simulations for the operating point with siginificant surface
velocities (Ief f = 200 A & ψ = −70◦ ) for the chosen
geometries are performed. The results are listed in Table VII.
It is shown, that sinusoidal field pole and notch in q-axis are
effective to reduce the surface velocity, and thus the acoustic
B. Analysis of the Entire Operating Area
TABLE VII: Simulation results Ief f = 200 A & ψ = −70◦
σDF T 2 (36, 0) Ẏ (36, 0) Mmean
Rotor alteration
Original
Sinusoidal Field Pole (SFP)
Notch in d-axis
Notch in q-axis
Combination SFP & q-notch
200
Mharm
Mmean
In Fig. 11 the characteristic torque-speed-diagram of the
36th torque harmonic and the radial force density wave order
(36, 0) are shown. Due to the change of reluctance, the maximum torque of the IPMSM is reduced by 2.5 Nm. The 36th
torque harmonic at its shifted rated speed n = 3600 min−1
is 11.27 Nm, which represents 7.4 % of the mean torque. The
maximum value of the radial force density order (36, 0) is
15 kN/m2 at n = 9600 min−1 and M = 65 Nm.
[%]
[kN/m2 ]
[mm/s]
[Nm]
36
72
36
20
34
17
14
3.1
1.7
2.9
1.5
1.2
83.4
76.3
78.2
66.0
70.4
30.0
25.2
39.1
11.0
13.3
4.9
1.3
5.8
4.0
1.3
Torque harmonic [Nm], temporal order s = 36
Torque M [Nm]
160
10
120
100
80
5
60
VI.
15
180
140
emission. Due to change of the reluctance, the effective torque
of these variants is reduced, particularly of the rotor with notch
in q-axis. However, the torque harmonics are also decreased.
The notch in d-axis doesn’t have considerable advantage to
torque and noise production. The combination of sinusoidal
field pole and notch in q-axis shows the best noise emission
and proportion of mean torque and its harmonics. This geometry will be examined for the entire operating area.
It is shown, that through the alteration of the rotor surface,
the harmonics of the air gap flux density can be reduced. This
correlates with the distribution of the force densities acting on
the stator teeth, which are responsible for the noise emission.
The noise reduction is obtained by the reduction of the relevant
force density order and thus the related surface velocities. The
combination of sinusoidal field pole and notch in q-axis shows
the most promising results. In this case, the relevant torque
harmonic 36th and radial flux density order (36, 0) are reduced
at least by 58 %.
R EFERENCES
40
[1]
20
0
0
5000
10000
15000
Speed n [min−1 ]
Force density [kN/m2 ], (s, r) = (36, 0)
16
180
14
160
Torque M [Nm]
[2]
0
200
[3]
[4]
[5]
[6]
12
140
[7]
120
10
100
8
80
6
[8]
[9]
60
4
40
[10]
2
20
0
0
5000
10000
Speed n [min−1 ]
15000
0
[11]
Fig. 11: Characteristic torque-speed-diagram of the 36th torque harmonic
(top) and the force density order (temporal s, spatial r) = (36, 0) (bottom).
1365
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C ONCLUSIONS
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